11.3.3 Mohr-Coulomb plasticity.doc

ABAQUS Analysis Users Manual 11.3.3Mohr-Coulomb plasticityProduct: ABAQUS/StandardMohr-Coulomb plasticity is defined by using the *MOHR COULOMB option together with the *MOHR COULOMB HARDENING option.References *MOHR COULOMB *MOHR COULOMB HARDENING “Material library: overview,” Section 9.1.1 “Inelastic behavior,” Section 11.1.1OverviewThe Mohr-Coulomb plasticity model: is used to model materials with the classical Mohr-Coloumb yield criterion; 用于模拟具有典型的莫尔库仑屈服准则的材料 allows the material to harden and/or soften isotropically; 容许材料等向硬化和（或）软化 uses a smooth flow potential that has a hyperbolic shape in the meridional stress 使用光滑流势，在子午面上为双曲线形 plane and a piecewise elliptic shape in the deviatoric stress plane; 在偏应力面上为平面和分段椭圆形 is used with the linear elastic material model (“Linear elastic behavior,” Section 10.2.1); and 用于线弹性材料； can be used for design applications in the geotechnical engineering area to simulate material response under essentially monotonic loading. 可以用于地质工程中的设计，用来模拟材料在常量荷载下的相应Elastic and plastic behavior 弹性和塑性行为The elastic part of the response is specified on the *ELASTIC option. Linear isotropic elasticity is assumed.弹性响应部分由*ELASTIC选项来指定。假定为线性等向弹性。The hardening behavior of the material is specified using the *MOHR COULOMB HARDENING option. Isotropic cohesion hardening is assumed. The hardening curve must describe the cohesion yield stress as a function of plastic strain and, possibly, temperature and predefined field variables. In defining this dependence at finite strains, “true” (Cauchy) stress and logarithmic strain values should be given.材料的硬化行为通过*MOHR COULOMH HARDENING选项来指定，假定为等向粘聚硬化。硬化曲线必须将粘聚屈服应力表达为塑性应变和（可能的话）温度及预设定场值的函数。在确定有限应变的可靠性时，必须指定真应力（柯西应力）和有效应变值。Rate dependency effects are not accounted for in this plasticity model.Usage:*MOHR COULOMB HARDENINGYield criterion 屈服准则The Mohr-Coulomb criterion assumes that failure occurs when the shear stress on any point in a material reaches a value that depends linearly on the normal stress in the same plane. The Mohr-Coulomb model is based on plotting Mohrs circle for states of stress at failure in the plane of the maximum and minimum principal stresses. The failure line is the best straight line that touches these Mohrs circles (Figure 11.3.31). 莫尔库仑准则假定破坏发生在材料中任意点的剪应力某一值，该值线性依赖于在该面的法向应力。莫尔库仑模式基于莫尔圆来判断最大和最小主应力平面上的破坏应力状态。破坏线是与莫尔圆相切的直线。Figure 11.3.31 Mohr-Coulomb failure model.Therefore, the Mohr-Coulomb model is defined by where is negative in compression. From Mohrs circle, Substituting for and , multiplying both sides by , and reducing, the Mohr-Coulomb model can be written as where is half of the difference between the maximum principal stress, , and the minimum principal stress, (and is, therefore, the maximum shear stress), is the average of the maximum and minimum principal stresses, and is the friction angle. For general states of stress the model is more conveniently written in terms of three stress invariants as where is the slope of the Mohr-Coulomb yield surface in the stress plane (see Figure 11.3.32), which is commonly referred to as the friction angle of the material and can depend on temperature and predefined field variables; is the cohesion of the material; andis the deviatoric polar angle defined as and is the equivalent pressure stress,is the Mises equivalent stress,is the third invariant of deviatoric stress,is the deviatoric stress.The friction angle controls the shape of the yield surface in the deviatoric plane as shown in Figure 11.3.32. The friction angle can range from . In the case of the Mohr-Coulomb model reduces to the pressure-independent Tresca model with a perfectly hexagonal deviatoric section. In the case of the Mohr-Coulomb model reduces to the “tension cut-off” Rankine model with a triangular deviatoric section and (this limiting case is not permitted within the Mohr-Coulomb model described here).While using one-element tests to verify the calibration of the model, it should be noted that the ABAQUS/Standard output variables SP1, SP2, and SP3 correspond to the principal stresses , , and , respectively.因此莫尔库仑准则定义为：其中在压缩时为负从莫尔圆上可知：两倍同时乘以并相减消掉and ，则莫尔库仑模式可以写成式中：为最大主应力和最小主应力之差的一半（即为最大剪应力）。最大和最小主应力的平均值，为内摩擦角。对于一般的情形应力模式通常用三个应力偏量来表示：式中; 为应力平面内莫尔库仑屈服面的斜率（图11.3.3-2），通常为材料的内幕擦角并且可以依赖于稳定和预定场值。为材料的粘聚力；为deviatoric polar angle定义为；为等效压应力；米斯等效压应力；应力偏量不变量；为偏应力。内摩擦角控制了偏平面中屈服面的形状，见图图11.3.3-2。内摩擦角在。当时莫尔库仑模式变为Tresca模式。在时莫尔库仑模式变为“tension cut-off”的Rankine模式，此时。当使用某一单元试验来校验模型准则时，应该注意ABAQUS/Standard的输出变量SP1,SP2和SP3分别于主应力, , and 相应。Figure 11.3.32 Mohr-Coulomb yield surface in meridional and deviatoric planes.Flow potential 流势The flow potential is chosen as a hyperbolic function in the meridional stress plane and the smooth elliptic function proposed by Mentrey and Willam (1995) in the deviatoric stress plane: where and is the dilation angle measured in the plane at high confining pressure and can depend on temperature and predefined field variables;is the initial cohesion yield stress, ;is the deviatoric polar angle defined previously;is a parameter, referred to as the meridional eccentricity, that defines the rate at which the hyperbolic function approaches the asymptote (the flow potential tends to a straight line in the meridional stress plane as the meridional eccentricity tends to zero); andis a parameter, referred to as the deviatoric eccentricity, that describes the “out-of-roundedness” of the deviatoric section in terms of the ratio between the shear stress along the extension meridian () and the shear stress along the compression meridian ().A default value of is provided for the meridional eccentricity, . By default, the deviatoric eccentricity, , is calculated as where is the Mohr-Coulomb friction angle; this calculation corresponds to matching the flow potential to the yield surface in both triaxial tension and compression in the deviatoric plane. Alternatively, ABAQUS/Standard allows the user to consider this deviatoric eccentricity as an independent material parameter; in this case the user provides its value directly. Convexity and smoothness of the elliptic function requires that . The upper limit, (or when the value of is not entered by the user), leads to , which describes the Mises circle in the deviatoric plane. The lower limit, (or when the value of is not entered by the user), leads to and would describe the Rankine triangle in the deviatoric plane (this limiting case is not permitted within the Mohr-Coulomb model described here). This flow potential, which is continuous and smooth, ensures that the flow direction is always uniquely defined. A family of hyperbolic potentials in the meridional stress plane is shown in Figure 11.3.33, and the flow potential in the deviatoric stress plane is shown in Figure 11.3.34.流势在应力子午面内采用双曲线函数，Mentrey and Willam (1995)建议在偏应力平面中采用光滑的椭圆函数：式中为膨胀角，可以在高限制压力下的平面内测定，并依赖于温度和预定场量；为初始屈服应力，为前面定义的偏极角 is a parameter, referred to as the meridional eccentricity, that defines the rate at which the hyperbolic function approaches the asymptote (the flow potential tends to a straight line in the meridional stress plane as the meridional eccentricity tends to zero); andis a parameter, referred to as the deviatoric eccentricity, that describes the “out-of-roundedness” of the deviatoric section in terms of the ratio between the shear stress along the extension meridian () and the shear stress along the compression meridian ().子午面离心率的默认值为0.1缺省情况下偏平面离心率定义为式中为莫尔库仑内摩擦角；这个计算与偏平面内的三轴拉伸和压缩时的屈服面的流势匹配。ABAQUS/Standard，也允许用户将该偏离心率独立于材料参数；在这种情况西用户直接将给值提供给ABAQUS。凸面及光滑的椭圆函数需要满足。上限（或当用户不输入值时），导致，表示了偏平面内的米斯圆。下限（或者当用户不输入值时），导致，表示了偏平面的Rankine三角形（这种情况在莫尔 库仑模式中是不允许的）。这个连续而光滑的流势保证了流向总是为唯一确定的。图11.3.3-3显示了材料应力平面内的一族双曲线势，图11.3.3-4显示了偏应力平面内的流势。Figure 11.3.33 Family of hyperbolic flow potentials in the meridional stress plane.Figure 11.3.34 Mentrey-Willam flow potential in the deviatoric stress plane.Flow in the meridional stress plane can be close to associated when the angle of friction, , and the angle of dilation, , are equal and the meridional eccentricity, , is very small; however, flow in this plane is, in general, nonassociated. Flow in the deviatoric stress plane is always nonassociated.Usage: Use either of the following options: *MOHR COULOMB*MOHR COULOMB, DEVIATORIC ECCENTRICITY=Nonassociated flowSince the plastic flow is n